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Get the formula between the
derivative in the direction of
the vector a
u  u
u
u
 cos   cos   cos 
a x
y
z

and the gradient
u
u
u
grad(u ) 
i
j
k
...

x
y
z

Thus, the directional
derivative is the scalar
product of the unit vector of
the direction vector and the
gradient (theorem 1):
u
 e a  grad(u )
a

Given the definition of the
scalar product, we get:
u
 grad(u ) cos ,
a

where  - angle between
vectors

a and grad(u)
...


Thus, the gradient of the
function indicates the
direction of the fastest
increase of the function, and
its modulus indicates the
highest rate of change of the
function (Theorem 3)
...


Hamilton William, 1805-1865,
Irish mathematician,
mechanic
...
In 13
years, he knew 12 languages
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x
y
z

Gradient vector properties:
1)
...
(C  u)  C  u

3)
...
  
2
v
 v

Example
Find the derivative in the
direction of the gradient of the
field function

u x y3
2

at point М(0;1;1)
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